Do US firms hire LLMs to their US offices? Or just JDs?

They also hire LLM graduates from other countries.

Yes they do, but is not as easy as it used to be (not that has even been really easy, but compared with 5 years ago getting an offer is now more difficult).

I believe its very important to work in an important law firm from the student country of origin, in order to obtain good recommendations that can open doors in US top law firms.